ANKARA/JERUSALEM, Jan 13 (Zerin Elci and Allyn Fisher-Ilan/Reuters) - Israel apologised to Turkey on Wednesday for publicly dressing down Ankara's ambassador in a dispute that has strained the once good ties between the Jewish state and the Muslim regional power.

Turkey had demanded a formal apology for Ambassador Oguz Celikkol's treatment on Monday and threatened to recall him.

But after receiving the letter of apology on Wednesday, Turkish Prime Minister Tayyip Erdogan threw a new barb at Israel, saying it should do more for peace in the region.

Turkey, as a Muslim country, is an important ally of Israel and in the past has helped forge contacts between the Jewish state and the Arab world.

But relations have deteriorated following criticism by Erdogan of Israel's offensive in the Gaza Strip last year.

The latest row broke out after Israeli Deputy Foreign Minister Danny Ayalon summoned Celikkol on Monday to protest against a Turkish television drama that portrayed Israeli diplomats as masterminds of a child abduction ring.

Ayalon invited media crews to the beginning of the meeting in Jerusalem and pointed out there was no Turkish flag on the table. He also said he was deliberately avoiding a handshake with the ambassador.

In television images broadcast in Turkey, Celikkol was seen seated on a low couch, accentuating the sense of a slight.

Ayalon later conceded his behaviour toward the envoy had been inappropriate. But Turkish President Abdullah Gul, who is scheduled to host Israeli Defence Minister Ehud Barak on Sunday, said that was insufficient and demanded a full apology.

Israel sent a formal letter of apology to Celikkol on Thursday.

I had no intention to humiliate you personally and apologize for the way the demarche was handled and perceived, Ayalon said in the letter, released by the Israeli government.

Please convey this to the Turkish people for whom we have great respect. I hope that both Israel and Turkey will seek diplomatic and courteous channels to convey messages as two allies should.

In response, Erdogan said the Turkish foreign ministry had received the expected, desired answer.

But he added more criticism of Israel, telling a news conference: Israel must put itself in order and it must be more just and more on the side of peace in the region.

PROTEST REMAINS VALID

Ayalon had said earlier that his protest against the Turkish criticism of Israel remained valid. Israeli Prime Minister Benjamim Netanyahu also said thought the protest was correct but handled badly, according to his office.

As a predominantly Muslim nation, albeit with a secular constitution, as well as a NATO military power, Turkey is a key ally for Israel in the Middle East. As well as providing security cooperation, Ankara has offered Israel diplomatic help in the past, notably mediating with Syria in 2008.

But ties have become frosty since Israel's war in the Palestinian Islamist-ruled Gaza Strip a year ago, which drew frequent public censure from Erdogan, whose AK Party's roots lie in political Islam.

Netanyahu has said Turkey was aligning itself with Muslim countries hostile to Israel like Iran since before the Gaza war.

There was similar outrage last year over a Turkish series which featured Israeli soldiers murdering Palestinian children.

On Tuesday in London, Turkish Foreign Minister Ahmet Davutoglu renewed his country's criticism of Israel over Gaza.

He said its 2008 invasion of the territory had marked the turning point in Turkish-Israeli relations.

Despite the row, a Turkish delegation is currently in Israel to wrap up the purchase of 10 Heron drones in a deal worth $180 million, Turkish defence officials said.

(Additional reporting by Alastair Macdonald and Dan Williams in Jerusalem, Darren Butler in Ankara, and Michele Kambas in Nicosia; Editing by Angus MacSwan)